Integrand size = 23, antiderivative size = 90 \[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\frac {2}{a c^2 (1+n) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}-\frac {2 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{a^{3/2} c^2 (1+n) \sqrt {c x}} \] Output:
2/a/c^2/(1+n)/(c*x)^(1/2)/(a/x+b*x^n)^(1/2)-2*x^(1/2)*arctanh(a^(1/2)/x^(1 /2)/(a/x+b*x^n)^(1/2))/a^(3/2)/c^2/(1+n)/(c*x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\frac {2 \left (\sqrt {a}-\sqrt {a+b x^{1+n}} \text {arctanh}\left (\frac {\sqrt {a+b x^{1+n}}}{\sqrt {a}}\right )\right )}{a^{3/2} c^2 (1+n) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}} \] Input:
Integrate[1/((c*x)^(5/2)*(a/x + b*x^n)^(3/2)),x]
Output:
(2*(Sqrt[a] - Sqrt[a + b*x^(1 + n)]*ArcTanh[Sqrt[a + b*x^(1 + n)]/Sqrt[a]] ))/(a^(3/2)*c^2*(1 + n)*Sqrt[c*x]*Sqrt[a/x + b*x^n])
Time = 0.43 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1936, 1937, 1935, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1936 |
\(\displaystyle \frac {\int \frac {1}{(c x)^{3/2} \sqrt {b x^n+\frac {a}{x}}}dx}{a c}+\frac {2}{a c^2 (n+1) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}\) |
\(\Big \downarrow \) 1937 |
\(\displaystyle \frac {\sqrt {x} \int \frac {1}{x^{3/2} \sqrt {b x^n+\frac {a}{x}}}dx}{a c^2 \sqrt {c x}}+\frac {2}{a c^2 (n+1) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {2}{a c^2 (n+1) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}-\frac {2 \sqrt {x} \int \frac {1}{1-\frac {a}{x \left (b x^n+\frac {a}{x}\right )}}d\frac {1}{\sqrt {x} \sqrt {b x^n+\frac {a}{x}}}}{a c^2 (n+1) \sqrt {c x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{a c^2 (n+1) \sqrt {c x} \sqrt {\frac {a}{x}+b x^n}}-\frac {2 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{a^{3/2} c^2 (n+1) \sqrt {c x}}\) |
Input:
Int[1/((c*x)^(5/2)*(a/x + b*x^n)^(3/2)),x]
Output:
2/(a*c^2*(1 + n)*Sqrt[c*x]*Sqrt[a/x + b*x^n]) - (2*Sqrt[x]*ArcTanh[Sqrt[a] /(Sqrt[x]*Sqrt[a/x + b*x^n])])/(a^(3/2)*c^2*(1 + n)*Sqrt[c*x])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n} , x] && ILtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && ( IntegerQ[j] || GtQ[c, 0])
Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a*x^j + b *x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]
\[\int \frac {1}{\left (c x \right )^{\frac {5}{2}} \left (\frac {a}{x}+b \,x^{n}\right )^{\frac {3}{2}}}d x\]
Input:
int(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x)
Output:
int(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x)
Exception generated. \[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\int \frac {1}{\left (c x\right )^{\frac {5}{2}} \left (\frac {a}{x} + b x^{n}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(c*x)**(5/2)/(a/x+b*x**n)**(3/2),x)
Output:
Integral(1/((c*x)**(5/2)*(a/x + b*x**n)**(3/2)), x)
\[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + \frac {a}{x}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^n + a/x)^(3/2)*(c*x)^(5/2)), x)
\[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + \frac {a}{x}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^n + a/x)^(3/2)*(c*x)^(5/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{5/2}\,{\left (b\,x^n+\frac {a}{x}\right )}^{3/2}} \,d x \] Input:
int(1/((c*x)^(5/2)*(b*x^n + a/x)^(3/2)),x)
Output:
int(1/((c*x)^(5/2)*(b*x^n + a/x)^(3/2)), x)
\[ \int \frac {1}{(c x)^{5/2} \left (\frac {a}{x}+b x^n\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x^{n} b x +a}}{x^{2 n} b^{2} x^{3}+2 x^{n} a b \,x^{2}+a^{2} x}d x \right )}{c^{3}} \] Input:
int(1/(c*x)^(5/2)/(a/x+b*x^n)^(3/2),x)
Output:
(sqrt(c)*int(sqrt(x**n*b*x + a)/(x**(2*n)*b**2*x**3 + 2*x**n*a*b*x**2 + a* *2*x),x))/c**3